Probability - Exponential Distribution

In summary, to derive the pdf of Y=(alpha)X from X~Exponential(alpha), you will need to find the cdf of X, use the relation Y=\alpha X to derive the cdf of Y, calculate the final probability using the cdf of X, and finally differentiate the cdf of Y to obtain its pdf. The steps are outlined as finding the cdf of X, using the relation to derive the cdf of Y, calculating the final probability, and differentiating the cdf to obtain the pdf.
  • #1
Brains_Tom
6
0
Here's the evil question:

Let X~Exponential(alpha). Derive and name the pdf of Y=(alpha)X
 
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  • #2
Hi! You should show some of your thoughts or working in your post...

But anyway, here are some steps to guide you along.

Step 1: Find the cumulative distribution function (cdf) of X. Since X is continuous, you will need to integrate the pdf of X [tex](f(x)=\alpha e^{-\alpha x}, for x\geq 0)[/tex] , with the lower limit being 0 (since we define [tex]x\geq0[/tex] for an exponential distribution) and the upper limit an arbitrary constant x.

Step 2: Use the relation [tex]Y= \alpha X[/tex] to derive the cdf of Y from the cdf of X. So [tex]F(y) = P(Y\leq y) = P(\alpha X\leq y) = P(X\leq \frac{y}{\alpha})[/tex]

Step 3: We can calculate this final probability since we know the cdf of X.

Step 4: Finally, differentiate the cdf of Y to obtain its pdf.

You will get a nice answer in the end.

All the best!

Note: Letters in small casing (e.g. x, y) represent constants while block letters (e.g. X, Y) are used to define the random variables.
 
Last edited:

What is the Exponential Distribution?

The Exponential Distribution is a probability distribution that models the time between events that occur independently at a constant rate. It is often used to model the waiting time for a specific event to occur.

What are the parameters of the Exponential Distribution?

The Exponential Distribution has one parameter, λ (lambda), which represents the rate at which events occur. This parameter determines the shape of the distribution and can take any positive value.

What is the probability density function (PDF) for the Exponential Distribution?

The probability density function (PDF) for the Exponential Distribution is f(x) = λe^-λx, where x is the time between events. This function describes the relative likelihood of observing a specific value for x.

How is the Exponential Distribution different from other probability distributions?

The Exponential Distribution is unique in that it is the only continuous probability distribution that has a constant failure rate. This means that the probability of an event occurring in a specific time interval is independent of the length of the interval.

How is the Exponential Distribution used in real-world applications?

The Exponential Distribution is commonly used in reliability and survival analysis, as well as in queuing theory and finance. It can also be used to model the time between earthquakes, radioactive decay, and the length of phone calls.

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